We give a numerical characterization of mutual orthogonality (that is,
complementarity) for subalgebras. In order to give such a characterization for
mutually orthogonal subalgebras $A$ and $B$ of the $k \times k$ matrix algebra
$M_k(\mathbb{C})$, where $A$ and $B$ are isomorphic to some $M_n(\mathbb{C})$
$(n \leq k)$, we consider a density matrix which is induced from the pair $\{A,
B\}$. We show that $A$ and $B$ are mutually orthogonal if and only if the von
Neumann entropy of the density matrix is the maximum value $2\log n$, which is
the logarithm of the dimension of the subfactors.